I’ve heard this quoted in many sermons (sometimes with “drown” swapped for “swim”), and I’ve seen it quoted in many books (sometimes in commentaries on John’s Gospel). It vividly illustrates that some parts of the Bible are simple and straightforward and that others require the highest level of historical and theological integration. Most people, however, quote this without attributing a source (e.g., “it has been said that”). Others attribute it (almost always without documentation) to Augustine, Gregory the Great, or John Owen.

In a famous image, Gregory explains that the Bible is like a river in which a lamb can walk and an elephant swim.

Gp ad Leand. 4. This memorable image gains in force because Gregory again uses sounds, in this case a sequence of vowels, to emphasise the point: agnus ambulet et elephas natet. This expression occurs in a letter to a distant friend, rather than in a work originally delivered orally, suggesting that Gregory expected even his written words to be apprehended aurally.

I have read someone who attributed the saying to Chrysostom’s comment on John’s Gospel. However, your research agrees with that of Craig Koester. I will need to correct my reference when I use the imagery. Thanks.

Joseph Hall used a similar phrase in the 17th century, but as a rebuke for the unlearned lay-people to stay close to shore & leave the deep waters for the elephants like himself.

XLIV In the waters of life, the divine scriptures, there are shallows, and there are deeps: shallows, where the lamb may wade; and deeps, where the elephant may swim. If we be not wise to distinguish, we may easily miscarry: he that can wade over the ford, cannot swim through the deep; and if he mistake the passage, he drowns. What infinite mischief hath arisen to the Church of God from the presumption of ignorant and unlettered men, that have taken upon them to interpret the most obscure scriptures, and pertinaciously defended their own sense!

[…] It has been said that the Bible is like a deep, broad body of water, shallow enough for a lamb to wade but deep enough for an elephant swim. Some parts are simple and straightforward, and others are not. Romans 9–11 is not. It is logically dense and theologically weighty. And it ends with an often-overlooked typological connection to Isaiah and Job. […]